Characterizations and construction methods for linear functional-repair storage codes

TL;DR

This paper characterizes linear functional-repair storage codes using admissible states of vector spaces, providing new construction methods, including symmetry-based approaches, to develop codes with optimal or improved parameters.

Contribution

It offers a precise characterization of linear functional-repair storage codes and introduces symmetry-based construction methods for designing codes with desired parameters.

Findings

Constructed codes meeting the cutset bound outside MBR and MSR points.

Developed a symmetry-based method to find and improve functional repair codes.

Discovered a code with parameters describable by only eight vector spaces.

Abstract

We present a precise characterization of linear functional-repair storage codes in terms of {\em admissible states/}, with each state made up from a collection of vector spaces over some fixed finite field. To illustrate the usefulness of our characterization, we provide several applications. We first describe a simple construction of functional-repair storage codes for a family of code parameters meeting the cutset bound outside the MBR and MSR points; these codes are conjectured to have optimal rate with respect to their repair locality. Then, we employ our characterization to develop a construction method to obtain functional repair codes for given parameters using symmetry groups, which can be used both to find new codes and to improve known ones. As an example of the latter use, we describe a beautiful functional-repair storage code that was found by this method, with parameters belonging to the family investigated earlier, which can be specified in terms of only eight different vector spaces.